Normal forms of differential systems

报告题目：Normal forms of differential systems

报告人：Valery Romanovski（教授博导），University of Maribor

报告时间：2024年9月28日（周六）14:00-16:00

报告地点：明理楼C302

报告人简介：

Valery Romanovski 现任斯洛文尼亚马里博尔(Maribor)大学教授, 应用数学与理论物理中心研究员，上海师范大学特聘教授。 于1986年列宁格勒州立大学(现圣彼得堡国立大学)获得物理与数学科学专业哲学博士，2001年白俄罗斯国家科学院数学研究所获物理与数学科学专业博士，主要研究方向为微分方程。Valery Romanovski教授2011年获斯洛文尼亚科学研究杰出贡献奖， 主持并参与多项国际科研项目，如2017斯洛文尼亚-匈牙利“微分方程中代数方法的应用”项目任首席研究员；2004、2006、2009、2012、2015斯洛文尼亚-美国双边项目任首席研究员；2008、20010、2011、2016斯洛文尼亚-俄罗斯双边项目任首席研究员等，并担任“Qualitative Theory of Dynamical Systems”、“Journal of Applied Analysis and Computation”等SCI杂志的编委，多次担任国际会议和讲习班的主讲人，组织过多次国际会议和讲习班。

报告摘要：There are two ways to compute Poincaré-Dulac normal forms of systems of ODEs. Under the original approach used by Poincare and Dulac the normalizing transformation is explicitly computed. On each step, the normalizing procedure requires the substitution of a polynomial to a series. Under the other approach, a normal form is computed using Lie transformations. In this case, the changes of coordinates are performed as actions of certain infinitesimal generators. In both cases, on each step the homological equation is solved in the vector space of polynomial vector fields where each component of the vector field is a homogeneous polynomial. We present the third way of computing normal forms of polynomial systems of ODEs where the coefficients of all terms are parameters. It is shown that the space of the parameters is a kind of dual space and the computation of normal forms can be performed in the space of parameters treated as the space of generalized vector fields, which we call the semilattice vector fields. The approach provides a simple way to parallelize the normal form computations opening the way to compute normal forms up to higher order than under previously known two approaches.

主办单位：理学院、人工智能研究院、非线性动力系统研究所、

数理力学研究中心、科学技术发展研究院